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Polar Length and its Quantities

April 20, 2014

I was reading this paper Minimal surfaces bounded by elastic lines (L. Giomi and L. Mahadevan)

there were some sort of dimensional analysis result about buckling .. however the Flexural Rigidity Quantity appeared in the paper.

The Flexural Rigidity \alpha=EI  and I  is the second moment of inertia

the second moment of inertia is actually an interesting quantity .. I have spent some time to differentiate between moment of inertia and second moment of inertia

after careful consideration .. my naming choice settled on Mass Moment of Inertia for the first and Area Moment of Inertia for the second

  AreaMomentOfInertia M0L4(RL2PL2)T0I0O0N0J0 <m^4>
      MassMomentOfInertia M1L2(RL0PL2)T0I0O0N0J0 <kg.m^2>

revising the concept of differentiating between Torque and Work (and differentiating between two types of lengths)   I have decided to modify my naming a little either in the source code and also in my writings after all .

Normal Length  now is called  Regular Length   or RL

Radius Length now is called Polar Length  or  PL

so Polar Length is any length that comes from a pole   ..   and revising the angle definition again 

the angle is the ratio between length on circle segment over the radius length

\theta=\frac{RL}{PL} 

Angle M0L0(RL1PL-1)T0I0O0N0J0 <1>

you can notice that dimension representation now includes Polar and Regular lengths are now written between parenthesis.

so L0(RL1PL-1)   express that L0 = RL-PL   

for list of all quantities please refer to this link http://quantitysystem.org/Home/Quantities

This was my claim that we are treating angles in a wrong way {angles are not dimensionless number}  (although angles are not dimensionless in my claim .. but in the framework I made an exception that angles can be added to other dimensionless numbers until I find a solid proof of my conjecture)

so .. in the case of second moment of inertia (Area Moment of Inertia)   there were this fact that we are integrating the square of radius to the area of the shape.

Reviewing the wikipedia definition

The second moment of area for an arbitrary shape with respect to an arbitrary axis \BB is defined as

J_{BB} = \int_A {\rho}^2 \, \mathrm dA

   dA = Differential area of the arbitrary shape

   \rho= Distance from the axis BB to dA

this means that we need to express it actually with RL2 PL2 

and I defined it that way .. so when using the calculator to make a those quantities you should write

fm = 1 <kg.m!^2>

am = 1<m^2.m!^2>

the <m!>  is the unit of polar length.

you can’t sum Polar length and Regular length because dimensionaly they are not the same

 

Flexural Rigidity then can be found by    EI = 1<Pa> * 1<m^2.m!^2>

FlexuralRigidity M1L3(RL1PL2)T-2I0O0N0J0 <kg.m^3/s^2>

Another interesting quantity is the Curvature

Curvature M0L-1(RL0PL-1)T0I0O0N0J0 <1/m>

Curvature is the 1/Polar Length  

you can define Curvature in calculator  as    s = 20 <1/m!> 

 

finally .. enjoy these new polar quantities definitions 🙂

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